Research Problems in Function Theory — Problem 7.29

Definitions and mathematical statement

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, fix $\alpha\in(0,1]$, and set $R_z=\alpha(1-|z|)$. The supplied question assumes a continuous function $f$ and the condition \[ \int_{|\zeta-z|<R_z}f(\zeta)\,d\zeta=0 \qquad(z\in\mathbb D), \] and asks whether $f$ must be holomorphic on $\mathbb D$.

Problem (L. Zalcman), pending source conventions. Determine whether this one-radius integral condition forces analyticity, and how the answer changes when continuity is assumed only inside the disk.

The displayed notation does not define an integral: the region is an open disk but the differential is $d\zeta$, normally used for a contour integral. A Morera-type reading would replace the region by the positively oriented circle $|\zeta-z|=R_z$ and retain $d\zeta$; an area reading would retain the disk and use area measure $dA(\zeta)$. These are different conditions and neither is silently selected.

The source's first continuity assumption and its final only-in-$\mathbb D$ question are identical as printed. They may have lost a closed-disk distinction, but that correction is not supplied. This also matters when $\alpha=1$: the candidate circle can touch $\partial\mathbb D$, and at $z=0$ it is the full unit circle, where a function continuous only on the open disk is not defined. Thus the integral type and boundary hypothesis must be recovered before a complete implication can be stated. The supplied summary reports the problem open as of the 2018 edition, with no definitive resolution located.

Short English statement

Does a vanishing integral at one prescribed radius around each point force holomorphy? The source mixes disk and contour notation and loses the distinction between its two continuity assumptions.

Sources

• Hayman, Research Problems in Function Theory (2018), Problem 7.29, attributed to L. Zalcman, https://arxiv.org/abs/1809.07200; source-TeX extraction, accessed 2026-07-29. The 2018 update was retained for status triage.
• Local registry data/unsolvedmath/problems.json, permanent ID $2307029$, source code catalogue entry, Research Problems in Function Theory --- Problem 7.29. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications. No separate integral-condition theorem citation is supplied.